class TestStateSpace(unittest.TestCase):
"""Tests for the StateSpace class."""
def setUp(self):
"""Set up a MIMO system to test operations on."""
A = [[-3., 4., 2.], [-1., -3., 0.], [2., 5., 3.]]
B = [[1., 4.], [-3., -3.], [-2., 1.]]
C = [[4., 2., -3.], [1., 4., 3.]]
D = [[-2., 4.], [0., 1.]]
a = [[4., 1.], [2., -3]]
b = [[5., 2.], [-3., -3.]]
c = [[2., -4], [0., 1.]]
d = [[3., 2.], [1., -1.]]
self.sys1 = StateSpace(A, B, C, D)
self.sys2 = StateSpace(a, b, c, d)
def testPole(self):
"""Evaluate the poles of a MIMO system."""
p = np.sort(self.sys1.pole())
true_p = np.sort([3.34747678408874,
-3.17373839204437 + 1.47492908003839j,
-3.17373839204437 - 1.47492908003839j])
np.testing.assert_array_almost_equal(p, true_p)
def testZero(self):
"""Evaluate the zeros of a SISO system."""
sys = StateSpace(self.sys1.A, [[3.], [-2.], [4.]], [[-1., 3., 2.]], [[-4.]])
z = sys.zero()
np.testing.assert_array_almost_equal(z, [4.26864638637134,
-3.75932319318567 + 1.10087776649554j,
-3.75932319318567 - 1.10087776649554j])
def testAdd(self):
"""Add two MIMO systems."""
A = [[-3., 4., 2., 0., 0.], [-1., -3., 0., 0., 0.],
[2., 5., 3., 0., 0.], [0., 0., 0., 4., 1.], [0., 0., 0., 2., -3.]]
B = [[1., 4.], [-3., -3.], [-2., 1.], [5., 2.], [-3., -3.]]
C = [[4., 2., -3., 2., -4.], [1., 4., 3., 0., 1.]]
D = [[1., 6.], [1., 0.]]
sys = self.sys1 + self.sys2
np.testing.assert_array_almost_equal(sys.A, A)
np.testing.assert_array_almost_equal(sys.B, B)
np.testing.assert_array_almost_equal(sys.C, C)
np.testing.assert_array_almost_equal(sys.D, D)
def testSub(self):
"""Subtract two MIMO systems."""
A = [[-3., 4., 2., 0., 0.], [-1., -3., 0., 0., 0.],
[2., 5., 3., 0., 0.], [0., 0., 0., 4., 1.], [0., 0., 0., 2., -3.]]
B = [[1., 4.], [-3., -3.], [-2., 1.], [5., 2.], [-3., -3.]]
C = [[4., 2., -3., -2., 4.], [1., 4., 3., 0., -1.]]
D = [[-5., 2.], [-1., 2.]]
sys = self.sys1 - self.sys2
np.testing.assert_array_almost_equal(sys.A, A)
np.testing.assert_array_almost_equal(sys.B, B)
np.testing.assert_array_almost_equal(sys.C, C)
np.testing.assert_array_almost_equal(sys.D, D)
def testMul(self):
"""Multiply two MIMO systems."""
A = [[4., 1., 0., 0., 0.], [2., -3., 0., 0., 0.], [2., 0., -3., 4., 2.],
[-6., 9., -1., -3., 0.], [-4., 9., 2., 5., 3.]]
B = [[5., 2.], [-3., -3.], [7., -2.], [-12., -3.], [-5., -5.]]
C = [[-4., 12., 4., 2., -3.], [0., 1., 1., 4., 3.]]
D = [[-2., -8.], [1., -1.]]
sys = self.sys1 * self.sys2
np.testing.assert_array_almost_equal(sys.A, A)
np.testing.assert_array_almost_equal(sys.B, B)
np.testing.assert_array_almost_equal(sys.C, C)
np.testing.assert_array_almost_equal(sys.D, D)
def testEvalFr(self):
"""Evaluate the frequency response at one frequency."""
A = [[-2, 0.5], [0.5, -0.3]]
B = [[0.3, -1.3], [0.1, 0.]]
C = [[0., 0.1], [-0.3, -0.2]]
D = [[0., -0.8], [-0.3, 0.]]
sys = StateSpace(A, B, C, D)
resp = [[4.37636761487965e-05 - 0.0152297592997812j,
-0.792603938730853 + 0.0261706783369803j],
[-0.331544857768052 + 0.0576105032822757j,
0.128919037199125 - 0.143824945295405j]]
np.testing.assert_almost_equal(sys.evalfr(1.), resp)
@unittest.skipIf(not slycot_check(), "slycot not installed")
def testFreqResp(self):
"""Evaluate the frequency response at multiple frequencies."""
A = [[-2, 0.5], [0.5, -0.3]]
B = [[0.3, -1.3], [0.1, 0.]]
C = [[0., 0.1], [-0.3, -0.2]]
D = [[0., -0.8], [-0.3, 0.]]
sys = StateSpace(A, B, C, D)
truemag = [[[0.0852992637230322, 0.00103596611395218],
[0.935374692849736, 0.799380720864549]],
[[0.55656854563842, 0.301542699860857],
[0.609178071542849, 0.0382108097985257]]]
truephase = [[[-0.566195599644593, -1.68063565332582],
[3.0465958317514, 3.14141384339534]],
[[2.90457947657161, 3.10601268291914],
[-0.438157380501337, -1.40720969147217]]]
trueomega = [0.1, 10.]
mag, phase, omega = sys.freqresp(trueomega)
np.testing.assert_almost_equal(mag, truemag)
np.testing.assert_almost_equal(phase, truephase)
np.testing.assert_equal(omega, trueomega)
@unittest.skipIf(not slycot_check(), "slycot not installed")
def testMinreal(self):
"""Test a minreal model reduction"""
#A = [-2, 0.5, 0; 0.5, -0.3, 0; 0, 0, -0.1]
A = [[-2, 0.5, 0], [0.5, -0.3, 0], [0, 0, -0.1]]
#B = [0.3, -1.3; 0.1, 0; 1, 0]
B = [[0.3, -1.3], [0.1, 0.], [1.0, 0.0]]
#C = [0, 0.1, 0; -0.3, -0.2, 0]
C = [[0., 0.1, 0.0], [-0.3, -0.2, 0.0]]
#D = [0 -0.8; -0.3 0]
D = [[0., -0.8], [-0.3, 0.]]
# sys = ss(A, B, C, D)
sys = StateSpace(A, B, C, D)
sysr = sys.minreal()
self.assertEqual(sysr.states, 2)
self.assertEqual(sysr.inputs, sys.inputs)
self.assertEqual(sysr.outputs, sys.outputs)
np.testing.assert_array_almost_equal(
eigvals(sysr.A), [-2.136154, -0.1638459])
def testAppendSS(self):
"""Test appending two state-space systems"""
A1 = [[-2, 0.5, 0], [0.5, -0.3, 0], [0, 0, -0.1]]
B1 = [[0.3, -1.3], [0.1, 0.], [1.0, 0.0]]
C1 = [[0., 0.1, 0.0], [-0.3, -0.2, 0.0]]
D1 = [[0., -0.8], [-0.3, 0.]]
A2 = [[-1.]]
B2 = [[1.2]]
C2 = [[0.5]]
D2 = [[0.4]]
A3 = [[-2, 0.5, 0, 0], [0.5, -0.3, 0, 0], [0, 0, -0.1, 0],
[0, 0, 0., -1.]]
B3 = [[0.3, -1.3, 0], [0.1, 0., 0], [1.0, 0.0, 0], [0., 0, 1.2]]
C3 = [[0., 0.1, 0.0, 0.0], [-0.3, -0.2, 0.0, 0.0], [0., 0., 0., 0.5]]
D3 = [[0., -0.8, 0.], [-0.3, 0., 0.], [0., 0., 0.4]]
sys1 = StateSpace(A1, B1, C1, D1)
sys2 = StateSpace(A2, B2, C2, D2)
sys3 = StateSpace(A3, B3, C3, D3)
sys3c = sys1.append(sys2)
np.testing.assert_array_almost_equal(sys3.A, sys3c.A)
np.testing.assert_array_almost_equal(sys3.B, sys3c.B)
np.testing.assert_array_almost_equal(sys3.C, sys3c.C)
np.testing.assert_array_almost_equal(sys3.D, sys3c.D)
def testAppendTF(self):
"""Test appending a state-space system with a tf"""
A1 = [[-2, 0.5, 0], [0.5, -0.3, 0], [0, 0, -0.1]]
B1 = [[0.3, -1.3], [0.1, 0.], [1.0, 0.0]]
C1 = [[0., 0.1, 0.0], [-0.3, -0.2, 0.0]]
D1 = [[0., -0.8], [-0.3, 0.]]
s = TransferFunction([1, 0], [1])
h = 1/(s+1)/(s+2)
sys1 = StateSpace(A1, B1, C1, D1)
sys2 = _convertToStateSpace(h)
sys3c = sys1.append(sys2)
np.testing.assert_array_almost_equal(sys1.A, sys3c.A[:3,:3])
np.testing.assert_array_almost_equal(sys1.B, sys3c.B[:3,:2])
np.testing.assert_array_almost_equal(sys1.C, sys3c.C[:2,:3])
np.testing.assert_array_almost_equal(sys1.D, sys3c.D[:2,:2])
np.testing.assert_array_almost_equal(sys2.A, sys3c.A[3:,3:])
np.testing.assert_array_almost_equal(sys2.B, sys3c.B[3:,2:])
np.testing.assert_array_almost_equal(sys2.C, sys3c.C[2:,3:])
np.testing.assert_array_almost_equal(sys2.D, sys3c.D[2:,2:])
np.testing.assert_array_almost_equal(sys3c.A[:3,3:], np.zeros( (3, 2)) )
np.testing.assert_array_almost_equal(sys3c.A[3:,:3], np.zeros( (2, 3)) )
def testArrayAccessSS(self):
sys1 = StateSpace([[1., 2.], [3., 4.]],
[[5., 6.], [6., 8.]],
[[9., 10.], [11., 12.]],
[[13., 14.], [15., 16.]], 1)
sys1_11 = sys1[0,1]
np.testing.assert_array_almost_equal(sys1_11.A,
sys1.A)
np.testing.assert_array_almost_equal(sys1_11.B,
sys1.B[:,1])
np.testing.assert_array_almost_equal(sys1_11.C,
sys1.C[0,:])
np.testing.assert_array_almost_equal(sys1_11.D,
sys1.D[0,1])
assert sys1.dt == sys1_11.dt
def test_dcgain_cont(self):
"""Test DC gain for continuous-time state-space systems"""
sys = StateSpace(-2.,6.,5.,0)
np.testing.assert_equal(sys.dcgain(), 15.)
sys2 = StateSpace(-2, [6., 4.], [[5.],[7.],[11]], np.zeros((3,2)))
expected = np.array([[15., 10.], [21., 14.], [33., 22.]])
np.testing.assert_array_equal(sys2.dcgain(), expected)
sys3 = StateSpace(0., 1., 1., 0.)
np.testing.assert_equal(sys3.dcgain(), np.nan)
def test_dcgain_discr(self):
"""Test DC gain for discrete-time state-space systems"""
# static gain
sys = StateSpace([], [], [], 2, True)
np.testing.assert_equal(sys.dcgain(), 2)
# averaging filter
sys = StateSpace(0.5, 0.5, 1, 0, True)
np.testing.assert_almost_equal(sys.dcgain(), 1)
# differencer
sys = StateSpace(0, 1, -1, 1, True)
np.testing.assert_equal(sys.dcgain(), 0)
# summer
sys = StateSpace(1, 1, 1, 0, True)
np.testing.assert_equal(sys.dcgain(), np.nan)
def test_dcgain_integrator(self):
"""DC gain when eigenvalue at DC returns appropriately sized array of nan"""
# the SISO case is also tested in test_dc_gain_{cont,discr}
import itertools
# iterate over input and output sizes, and continuous (dt=None) and discrete (dt=True) time
for inputs,outputs,dt in itertools.product(range(1,6),range(1,6),[None,True]):
states = max(inputs,outputs)
# a matrix that is singular at DC, and has no "useless" states as in _remove_useless_states
a = np.triu(np.tile(2,(states,states)))
# eigenvalues all +2, except for ...
a[0,0] = 0 if dt is None else 1
b = np.eye(max(inputs,states))[:states,:inputs]
c = np.eye(max(outputs,states))[:outputs,:states]
d = np.zeros((outputs,inputs))
sys = StateSpace(a,b,c,d,dt)
dc = np.squeeze(np.tile(np.nan,(outputs,inputs)))
np.testing.assert_array_equal(dc, sys.dcgain())
def test_scalarStaticGain(self):
"""Regression: can we create a scalar static gain?"""
g1=StateSpace([],[],[],[2])
g2=StateSpace([],[],[],[3])
# make sure StateSpace internals, specifically ABC matrix
# sizes, are OK for LTI operations
g3 = g1*g2
self.assertEqual(6, g3.D[0,0])
g4 = g1+g2
self.assertEqual(5, g4.D[0,0])
g5 = g1.feedback(g2)
self.assertAlmostEqual(2./7, g5.D[0,0])
g6 = g1.append(g2)
np.testing.assert_array_equal(np.diag([2,3]),g6.D)
def test_matrixStaticGain(self):
"""Regression: can we create matrix static gains?"""
d1 = np.matrix([[1,2,3],[4,5,6]])
d2 = np.matrix([[7,8],[9,10],[11,12]])
g1=StateSpace([],[],[],d1)
# _remove_useless_states was making A = [[0]]
self.assertEqual((0,0), g1.A.shape)
g2=StateSpace([],[],[],d2)
g3=StateSpace([],[],[],d2.T)
h1 = g1*g2
np.testing.assert_array_equal(d1*d2, h1.D)
h2 = g1+g3
np.testing.assert_array_equal(d1+d2.T, h2.D)
h3 = g1.feedback(g2)
np.testing.assert_array_almost_equal(solve(np.eye(2)+d1*d2,d1), h3.D)
h4 = g1.append(g2)
np.testing.assert_array_equal(block_diag(d1,d2),h4.D)
def test_remove_useless_states(self):
"""Regression: _remove_useless_states gives correct ABC sizes"""
g1 = StateSpace(np.zeros((3,3)),
np.zeros((3,4)),
np.zeros((5,3)),
np.zeros((5,4)))
self.assertEqual((0,0), g1.A.shape)
self.assertEqual((0,4), g1.B.shape)
self.assertEqual((5,0), g1.C.shape)
self.assertEqual((5,4), g1.D.shape)
self.assertEqual(0, g1.states)
def test_BadEmptyMatrices(self):
"""Mismatched ABCD matrices when some are empty"""
self.assertRaises(ValueError,StateSpace, [1], [], [], [1])
self.assertRaises(ValueError,StateSpace, [1], [1], [], [1])
self.assertRaises(ValueError,StateSpace, [1], [], [1], [1])
self.assertRaises(ValueError,StateSpace, [], [1], [], [1])
self.assertRaises(ValueError,StateSpace, [], [1], [1], [1])
self.assertRaises(ValueError,StateSpace, [], [], [1], [1])
self.assertRaises(ValueError,StateSpace, [1], [1], [1], [])
def test_minrealStaticGain(self):
"""Regression: minreal on static gain was failing"""
g1 = StateSpace([],[],[],[1])
g2 = g1.minreal()
np.testing.assert_array_equal(g1.A, g2.A)
np.testing.assert_array_equal(g1.B, g2.B)
np.testing.assert_array_equal(g1.C, g2.C)
np.testing.assert_array_equal(g1.D, g2.D)
class TestStateSpace(unittest.TestCase):
"""Tests for the StateSpace class."""
def setUp(self):
"""Set up a MIMO system to test operations on."""
use_numpy_matrix(False)
# sys1: 3-states square system (2 inputs x 2 outputs)
A322 = [[-3., 4., 2.], [-1., -3., 0.], [2., 5., 3.]]
B322 = [[1., 4.], [-3., -3.], [-2., 1.]]
C322 = [[4., 2., -3.], [1., 4., 3.]]
D322 = [[-2., 4.], [0., 1.]]
self.sys322 = StateSpace(A322, B322, C322, D322)
# sys1: 2-states square system (2 inputs x 2 outputs)
A222 = [[4., 1.], [2., -3]]
B222 = [[5., 2.], [-3., -3.]]
C222 = [[2., -4], [0., 1.]]
D222 = [[3., 2.], [1., -1.]]
self.sys222 = StateSpace(A222, B222, C222, D222)
# sys3: 6 states non square system (2 inputs x 3 outputs)
A623 = np.array([[1, 0, 0, 0, 0, 0], [0, 1, 0, 0, 0, 0],
[0, 0, 3, 0, 0, 0], [0, 0, 0, -4, 0, 0],
[0, 0, 0, 0, -1, 0], [0, 0, 0, 0, 0, 3]])
B623 = np.array([[0, -1], [-1, 0], [1, -1], [0, 0], [0, 1], [-1, -1]])
C623 = np.array([[1, 0, 0, 1, 0, 0], [0, 1, 0, 1, 0, 1],
[0, 0, 1, 0, 0, 1]])
D623 = np.zeros((3, 2))
self.sys623 = StateSpace(A623, B623, C623, D623)
def test_matlab_style_constructor(self):
# Use (deprecated?) matrix-style construction string (w/ warnings off)
import warnings
warnings.filterwarnings("ignore") # turn off warnings
sys = StateSpace("-1 1; 0 2", "0; 1", "1, 0", "0")
warnings.resetwarnings() # put things back to original state
self.assertEqual(sys.A.shape, (2, 2))
self.assertEqual(sys.B.shape, (2, 1))
self.assertEqual(sys.C.shape, (1, 2))
self.assertEqual(sys.D.shape, (1, 1))
if defaults['statesp.use_numpy_matrix']:
for X in [sys.A, sys.B, sys.C, sys.D]:
self.assertTrue(isinstance(X, np.matrix))
else:
for X in [sys.A, sys.B, sys.C, sys.D]:
self.assertTrue(isinstance(X, np.ndarray))
def test_pole(self):
"""Evaluate the poles of a MIMO system."""
p = np.sort(self.sys322.pole())
true_p = np.sort([
3.34747678408874, -3.17373839204437 + 1.47492908003839j,
-3.17373839204437 - 1.47492908003839j
])
np.testing.assert_array_almost_equal(p, true_p)
def test_zero_empty(self):
"""Test to make sure zero() works with no zeros in system."""
sys = _convertToStateSpace(TransferFunction([1], [1, 2, 1]))
np.testing.assert_array_equal(sys.zero(), np.array([]))
@unittest.skipIf(not slycot_check(), "slycot not installed")
def test_zero_siso(self):
"""Evaluate the zeros of a SISO system."""
# extract only first input / first output system of sys222. This system is denoted sys111
# or tf111
tf111 = ss2tf(self.sys222)
sys111 = tf2ss(tf111[0, 0])
# compute zeros as root of the characteristic polynomial at the numerator of tf111
# this method is simple and assumed as valid in this test
true_z = np.sort(tf111[0, 0].zero())
# Compute the zeros through ab08nd, which is tested here
z = np.sort(sys111.zero())
np.testing.assert_almost_equal(true_z, z)
@unittest.skipIf(not slycot_check(), "slycot not installed")
def test_zero_mimo_sys322_square(self):
"""Evaluate the zeros of a square MIMO system."""
z = np.sort(self.sys322.zero())
true_z = np.sort([44.41465, -0.490252, -5.924398])
np.testing.assert_array_almost_equal(z, true_z)
@unittest.skipIf(not slycot_check(), "slycot not installed")
def test_zero_mimo_sys222_square(self):
"""Evaluate the zeros of a square MIMO system."""
z = np.sort(self.sys222.zero())
true_z = np.sort([-10.568501, 3.368501])
np.testing.assert_array_almost_equal(z, true_z)
@unittest.skipIf(not slycot_check(), "slycot not installed")
def test_zero_mimo_sys623_non_square(self):
"""Evaluate the zeros of a non square MIMO system."""
z = np.sort(self.sys623.zero())
true_z = np.sort([2., -1.])
np.testing.assert_array_almost_equal(z, true_z)
def test_add_ss(self):
"""Add two MIMO systems."""
A = [[-3., 4., 2., 0., 0.], [-1., -3., 0., 0., 0.],
[2., 5., 3., 0., 0.], [0., 0., 0., 4., 1.], [0., 0., 0., 2., -3.]]
B = [[1., 4.], [-3., -3.], [-2., 1.], [5., 2.], [-3., -3.]]
C = [[4., 2., -3., 2., -4.], [1., 4., 3., 0., 1.]]
D = [[1., 6.], [1., 0.]]
sys = self.sys322 + self.sys222
np.testing.assert_array_almost_equal(sys.A, A)
np.testing.assert_array_almost_equal(sys.B, B)
np.testing.assert_array_almost_equal(sys.C, C)
np.testing.assert_array_almost_equal(sys.D, D)
def test_subtract_ss(self):
"""Subtract two MIMO systems."""
A = [[-3., 4., 2., 0., 0.], [-1., -3., 0., 0., 0.],
[2., 5., 3., 0., 0.], [0., 0., 0., 4., 1.], [0., 0., 0., 2., -3.]]
B = [[1., 4.], [-3., -3.], [-2., 1.], [5., 2.], [-3., -3.]]
C = [[4., 2., -3., -2., 4.], [1., 4., 3., 0., -1.]]
D = [[-5., 2.], [-1., 2.]]
sys = self.sys322 - self.sys222
np.testing.assert_array_almost_equal(sys.A, A)
np.testing.assert_array_almost_equal(sys.B, B)
np.testing.assert_array_almost_equal(sys.C, C)
np.testing.assert_array_almost_equal(sys.D, D)
def test_multiply_ss(self):
"""Multiply two MIMO systems."""
A = [[4., 1., 0., 0., 0.], [2., -3., 0., 0., 0.],
[2., 0., -3., 4., 2.], [-6., 9., -1., -3., 0.],
[-4., 9., 2., 5., 3.]]
B = [[5., 2.], [-3., -3.], [7., -2.], [-12., -3.], [-5., -5.]]
C = [[-4., 12., 4., 2., -3.], [0., 1., 1., 4., 3.]]
D = [[-2., -8.], [1., -1.]]
sys = self.sys322 * self.sys222
np.testing.assert_array_almost_equal(sys.A, A)
np.testing.assert_array_almost_equal(sys.B, B)
np.testing.assert_array_almost_equal(sys.C, C)
np.testing.assert_array_almost_equal(sys.D, D)
def test_evalfr(self):
"""Evaluate the frequency response at one frequency."""
A = [[-2, 0.5], [0.5, -0.3]]
B = [[0.3, -1.3], [0.1, 0.]]
C = [[0., 0.1], [-0.3, -0.2]]
D = [[0., -0.8], [-0.3, 0.]]
sys = StateSpace(A, B, C, D)
resp = [[
4.37636761487965e-05 - 0.0152297592997812j,
-0.792603938730853 + 0.0261706783369803j
],
[
-0.331544857768052 + 0.0576105032822757j,
0.128919037199125 - 0.143824945295405j
]]
# Correct versions of the call
np.testing.assert_almost_equal(evalfr(sys, 1j), resp)
np.testing.assert_almost_equal(sys._evalfr(1.), resp)
# Deprecated version of the call (should generate warning)
import warnings
with warnings.catch_warnings(record=True) as w:
# Set up warnings filter to only show warnings in control module
warnings.filterwarnings("ignore")
warnings.filterwarnings("always", module="control")
# Make sure that we get a pending deprecation warning
sys.evalfr(1.)
assert len(w) == 1
assert issubclass(w[-1].category, PendingDeprecationWarning)
# Leave the warnings filter like we found it
warnings.resetwarnings()
@unittest.skipIf(not slycot_check(), "slycot not installed")
def test_freq_resp(self):
"""Evaluate the frequency response at multiple frequencies."""
A = [[-2, 0.5], [0.5, -0.3]]
B = [[0.3, -1.3], [0.1, 0.]]
C = [[0., 0.1], [-0.3, -0.2]]
D = [[0., -0.8], [-0.3, 0.]]
sys = StateSpace(A, B, C, D)
true_mag = [[[0.0852992637230322, 0.00103596611395218],
[0.935374692849736, 0.799380720864549]],
[[0.55656854563842, 0.301542699860857],
[0.609178071542849, 0.0382108097985257]]]
true_phase = [[[-0.566195599644593, -1.68063565332582],
[3.0465958317514, 3.14141384339534]],
[[2.90457947657161, 3.10601268291914],
[-0.438157380501337, -1.40720969147217]]]
true_omega = [0.1, 10.]
mag, phase, omega = sys.freqresp(true_omega)
np.testing.assert_almost_equal(mag, true_mag)
np.testing.assert_almost_equal(phase, true_phase)
np.testing.assert_equal(omega, true_omega)
@unittest.skipIf(not slycot_check(), "slycot not installed")
def test_minreal(self):
"""Test a minreal model reduction."""
# A = [-2, 0.5, 0; 0.5, -0.3, 0; 0, 0, -0.1]
A = [[-2, 0.5, 0], [0.5, -0.3, 0], [0, 0, -0.1]]
# B = [0.3, -1.3; 0.1, 0; 1, 0]
B = [[0.3, -1.3], [0.1, 0.], [1.0, 0.0]]
# C = [0, 0.1, 0; -0.3, -0.2, 0]
C = [[0., 0.1, 0.0], [-0.3, -0.2, 0.0]]
# D = [0 -0.8; -0.3 0]
D = [[0., -0.8], [-0.3, 0.]]
# sys = ss(A, B, C, D)
sys = StateSpace(A, B, C, D)
sysr = sys.minreal()
self.assertEqual(sysr.states, 2)
self.assertEqual(sysr.inputs, sys.inputs)
self.assertEqual(sysr.outputs, sys.outputs)
np.testing.assert_array_almost_equal(eigvals(sysr.A),
[-2.136154, -0.1638459])
def test_append_ss(self):
"""Test appending two state-space systems."""
A1 = [[-2, 0.5, 0], [0.5, -0.3, 0], [0, 0, -0.1]]
B1 = [[0.3, -1.3], [0.1, 0.], [1.0, 0.0]]
C1 = [[0., 0.1, 0.0], [-0.3, -0.2, 0.0]]
D1 = [[0., -0.8], [-0.3, 0.]]
A2 = [[-1.]]
B2 = [[1.2]]
C2 = [[0.5]]
D2 = [[0.4]]
A3 = [[-2, 0.5, 0, 0], [0.5, -0.3, 0, 0], [0, 0, -0.1, 0],
[0, 0, 0., -1.]]
B3 = [[0.3, -1.3, 0], [0.1, 0., 0], [1.0, 0.0, 0], [0., 0, 1.2]]
C3 = [[0., 0.1, 0.0, 0.0], [-0.3, -0.2, 0.0, 0.0], [0., 0., 0., 0.5]]
D3 = [[0., -0.8, 0.], [-0.3, 0., 0.], [0., 0., 0.4]]
sys1 = StateSpace(A1, B1, C1, D1)
sys2 = StateSpace(A2, B2, C2, D2)
sys3 = StateSpace(A3, B3, C3, D3)
sys3c = sys1.append(sys2)
np.testing.assert_array_almost_equal(sys3.A, sys3c.A)
np.testing.assert_array_almost_equal(sys3.B, sys3c.B)
np.testing.assert_array_almost_equal(sys3.C, sys3c.C)
np.testing.assert_array_almost_equal(sys3.D, sys3c.D)
def test_append_tf(self):
"""Test appending a state-space system with a tf"""
A1 = [[-2, 0.5, 0], [0.5, -0.3, 0], [0, 0, -0.1]]
B1 = [[0.3, -1.3], [0.1, 0.], [1.0, 0.0]]
C1 = [[0., 0.1, 0.0], [-0.3, -0.2, 0.0]]
D1 = [[0., -0.8], [-0.3, 0.]]
s = TransferFunction([1, 0], [1])
h = 1 / (s + 1) / (s + 2)
sys1 = StateSpace(A1, B1, C1, D1)
sys2 = _convertToStateSpace(h)
sys3c = sys1.append(sys2)
np.testing.assert_array_almost_equal(sys1.A, sys3c.A[:3, :3])
np.testing.assert_array_almost_equal(sys1.B, sys3c.B[:3, :2])
np.testing.assert_array_almost_equal(sys1.C, sys3c.C[:2, :3])
np.testing.assert_array_almost_equal(sys1.D, sys3c.D[:2, :2])
np.testing.assert_array_almost_equal(sys2.A, sys3c.A[3:, 3:])
np.testing.assert_array_almost_equal(sys2.B, sys3c.B[3:, 2:])
np.testing.assert_array_almost_equal(sys2.C, sys3c.C[2:, 3:])
np.testing.assert_array_almost_equal(sys2.D, sys3c.D[2:, 2:])
np.testing.assert_array_almost_equal(sys3c.A[:3, 3:], np.zeros((3, 2)))
np.testing.assert_array_almost_equal(sys3c.A[3:, :3], np.zeros((2, 3)))
def test_array_access_ss(self):
sys1 = StateSpace([[1., 2.], [3., 4.]], [[5., 6.], [6., 8.]],
[[9., 10.], [11., 12.]], [[13., 14.], [15., 16.]], 1)
sys1_11 = sys1[0, 1]
np.testing.assert_array_almost_equal(sys1_11.A, sys1.A)
np.testing.assert_array_almost_equal(sys1_11.B, sys1.B[:, [1]])
np.testing.assert_array_almost_equal(sys1_11.C, sys1.C[[0], :])
np.testing.assert_array_almost_equal(sys1_11.D, sys1.D[0, 1])
assert sys1.dt == sys1_11.dt
def test_dc_gain_cont(self):
"""Test DC gain for continuous-time state-space systems."""
sys = StateSpace(-2., 6., 5., 0)
np.testing.assert_equal(sys.dcgain(), 15.)
sys2 = StateSpace(-2, [[6., 4.]], [[5.], [7.], [11]], np.zeros((3, 2)))
expected = np.array([[15., 10.], [21., 14.], [33., 22.]])
np.testing.assert_array_equal(sys2.dcgain(), expected)
sys3 = StateSpace(0., 1., 1., 0.)
np.testing.assert_equal(sys3.dcgain(), np.nan)
def test_dc_gain_discr(self):
"""Test DC gain for discrete-time state-space systems."""
# static gain
sys = StateSpace([], [], [], 2, True)
np.testing.assert_equal(sys.dcgain(), 2)
# averaging filter
sys = StateSpace(0.5, 0.5, 1, 0, True)
np.testing.assert_almost_equal(sys.dcgain(), 1)
# differencer
sys = StateSpace(0, 1, -1, 1, True)
np.testing.assert_equal(sys.dcgain(), 0)
# summer
sys = StateSpace(1, 1, 1, 0, True)
np.testing.assert_equal(sys.dcgain(), np.nan)
def test_dc_gain_integrator(self):
"""DC gain when eigenvalue at DC returns appropriately sized array of nan."""
# the SISO case is also tested in test_dc_gain_{cont,discr}
import itertools
# iterate over input and output sizes, and continuous (dt=None) and discrete (dt=True) time
for inputs, outputs, dt in itertools.product(range(1, 6), range(1, 6),
[None, True]):
states = max(inputs, outputs)
# a matrix that is singular at DC, and has no "useless" states as in
# _remove_useless_states
a = np.triu(np.tile(2, (states, states)))
# eigenvalues all +2, except for ...
a[0, 0] = 0 if dt is None else 1
b = np.eye(max(inputs, states))[:states, :inputs]
c = np.eye(max(outputs, states))[:outputs, :states]
d = np.zeros((outputs, inputs))
sys = StateSpace(a, b, c, d, dt)
dc = np.squeeze(np.tile(np.nan, (outputs, inputs)))
np.testing.assert_array_equal(dc, sys.dcgain())
def test_scalar_static_gain(self):
"""Regression: can we create a scalar static gain?"""
g1 = StateSpace([], [], [], [2])
g2 = StateSpace([], [], [], [3])
# make sure StateSpace internals, specifically ABC matrix
# sizes, are OK for LTI operations
g3 = g1 * g2
self.assertEqual(6, g3.D[0, 0])
g4 = g1 + g2
self.assertEqual(5, g4.D[0, 0])
g5 = g1.feedback(g2)
np.testing.assert_array_almost_equal(2. / 7, g5.D[0, 0])
g6 = g1.append(g2)
np.testing.assert_array_equal(np.diag([2, 3]), g6.D)
def test_matrix_static_gain(self):
"""Regression: can we create matrix static gains?"""
d1 = np.array([[1, 2, 3], [4, 5, 6]])
d2 = np.array([[7, 8], [9, 10], [11, 12]])
g1 = StateSpace([], [], [], d1)
# _remove_useless_states was making A = [[0]]
self.assertEqual((0, 0), g1.A.shape)
g2 = StateSpace([], [], [], d2)
g3 = StateSpace([], [], [], d2.T)
h1 = g1 * g2
np.testing.assert_array_equal(np.dot(d1, d2), h1.D)
h2 = g1 + g3
np.testing.assert_array_equal(d1 + d2.T, h2.D)
h3 = g1.feedback(g2)
np.testing.assert_array_almost_equal(
solve(np.eye(2) + np.dot(d1, d2), d1), h3.D)
h4 = g1.append(g2)
np.testing.assert_array_equal(block_diag(d1, d2), h4.D)
def test_remove_useless_states(self):
"""Regression: _remove_useless_states gives correct ABC sizes."""
g1 = StateSpace(np.zeros((3, 3)), np.zeros((3, 4)), np.zeros((5, 3)),
np.zeros((5, 4)))
self.assertEqual((0, 0), g1.A.shape)
self.assertEqual((0, 4), g1.B.shape)
self.assertEqual((5, 0), g1.C.shape)
self.assertEqual((5, 4), g1.D.shape)
self.assertEqual(0, g1.states)
def test_bad_empty_matrices(self):
"""Mismatched ABCD matrices when some are empty."""
self.assertRaises(ValueError, StateSpace, [1], [], [], [1])
self.assertRaises(ValueError, StateSpace, [1], [1], [], [1])
self.assertRaises(ValueError, StateSpace, [1], [], [1], [1])
self.assertRaises(ValueError, StateSpace, [], [1], [], [1])
self.assertRaises(ValueError, StateSpace, [], [1], [1], [1])
self.assertRaises(ValueError, StateSpace, [], [], [1], [1])
self.assertRaises(ValueError, StateSpace, [1], [1], [1], [])
def test_minreal_static_gain(self):
"""Regression: minreal on static gain was failing."""
g1 = StateSpace([], [], [], [1])
g2 = g1.minreal()
np.testing.assert_array_equal(g1.A, g2.A)
np.testing.assert_array_equal(g1.B, g2.B)
np.testing.assert_array_equal(g1.C, g2.C)
np.testing.assert_array_equal(g1.D, g2.D)
def test_empty(self):
"""Regression: can we create an empty StateSpace object?"""
g1 = StateSpace([], [], [], [])
self.assertEqual(0, g1.states)
self.assertEqual(0, g1.inputs)
self.assertEqual(0, g1.outputs)
def test_matrix_to_state_space(self):
"""_convertToStateSpace(matrix) gives ss([],[],[],D)"""
D = np.array([[1, 2, 3], [4, 5, 6]])
g = _convertToStateSpace(D)
def empty(shape):
m = np.array([])
m.shape = shape
return m
np.testing.assert_array_equal(empty((0, 0)), g.A)
np.testing.assert_array_equal(empty((0, D.shape[1])), g.B)
np.testing.assert_array_equal(empty((D.shape[0], 0)), g.C)
np.testing.assert_array_equal(D, g.D)
def test_lft(self):
""" test lft function with result obtained from matlab implementation"""
# test case
A = [[1, 2, 3], [1, 4, 5], [2, 3, 4]]
B = [[0, 2], [5, 6], [5, 2]]
C = [[1, 4, 5], [2, 3, 0]]
D = [[0, 0], [3, 0]]
P = StateSpace(A, B, C, D)
Ak = [[0, 2, 3], [2, 3, 5], [2, 1, 9]]
Bk = [[1, 1], [2, 3], [9, 4]]
Ck = [[1, 4, 5], [2, 3, 6]]
Dk = [[0, 2], [0, 0]]
K = StateSpace(Ak, Bk, Ck, Dk)
# case 1
pk = P.lft(K, 2, 1)
Amatlab = [
1, 2, 3, 4, 6, 12, 1, 4, 5, 17, 38, 61, 2, 3, 4, 9, 26, 37, 2, 3,
0, 3, 14, 18, 4, 6, 0, 8, 27, 35, 18, 27, 0, 29, 109, 144
]
Bmatlab = [0, 10, 10, 7, 15, 58]
Cmatlab = [1, 4, 5, 0, 0, 0]
Dmatlab = [0]
np.testing.assert_allclose(np.array(pk.A).reshape(-1), Amatlab)
np.testing.assert_allclose(np.array(pk.B).reshape(-1), Bmatlab)
np.testing.assert_allclose(np.array(pk.C).reshape(-1), Cmatlab)
np.testing.assert_allclose(np.array(pk.D).reshape(-1), Dmatlab)
# case 2
pk = P.lft(K)
Amatlab = [
1, 2, 3, 4, 6, 12, -3, -2, 5, 11, 14, 31, -2, -3, 4, 3, 2, 7, 0.6,
3.4, 5, -0.6, -0.4, 0, 0.8, 6.2, 10, 0.2, -4.2, -4, 7.4, 33.6, 45,
-0.4, -8.6, -3
]
Bmatlab = []
Cmatlab = []
Dmatlab = []
np.testing.assert_allclose(np.array(pk.A).reshape(-1), Amatlab)
np.testing.assert_allclose(np.array(pk.B).reshape(-1), Bmatlab)
np.testing.assert_allclose(np.array(pk.C).reshape(-1), Cmatlab)
np.testing.assert_allclose(np.array(pk.D).reshape(-1), Dmatlab)
def test_horner(self):
"""Test horner() function"""
# Make sure we can compute the transfer function at a complex value
self.sys322.horner(1. + 1.j)
# Make sure result agrees with frequency response
mag, phase, omega = self.sys322.freqresp([1])
np.testing.assert_array_almost_equal(
self.sys322.horner(1.j),
mag[:, :, 0] * np.exp(1.j * phase[:, :, 0]))
def tearDown(self):
reset_defaults() # reset configuration defaults
class TestStateSpace(unittest.TestCase):
"""Tests for the StateSpace class."""
def setUp(self):
"""Set up a MIMO system to test operations on."""
A = [[-3., 4., 2.], [-1., -3., 0.], [2., 5., 3.]]
B = [[1., 4.], [-3., -3.], [-2., 1.]]
C = [[4., 2., -3.], [1., 4., 3.]]
D = [[-2., 4.], [0., 1.]]
a = [[4., 1.], [2., -3]]
b = [[5., 2.], [-3., -3.]]
c = [[2., -4], [0., 1.]]
d = [[3., 2.], [1., -1.]]
self.sys1 = StateSpace(A, B, C, D)
self.sys2 = StateSpace(a, b, c, d)
def testPole(self):
"""Evaluate the poles of a MIMO system."""
p = np.sort(self.sys1.pole())
true_p = np.sort([
3.34747678408874, -3.17373839204437 + 1.47492908003839j,
-3.17373839204437 - 1.47492908003839j
])
np.testing.assert_array_almost_equal(p, true_p)
def testZero(self):
"""Evaluate the zeros of a SISO system."""
sys = StateSpace(self.sys1.A, [[3.], [-2.], [4.]], [[-1., 3., 2.]],
[[-4.]])
z = sys.zero()
np.testing.assert_array_almost_equal(z, [
4.26864638637134, -3.75932319318567 + 1.10087776649554j,
-3.75932319318567 - 1.10087776649554j
])
def testAdd(self):
"""Add two MIMO systems."""
A = [[-3., 4., 2., 0., 0.], [-1., -3., 0., 0., 0.],
[2., 5., 3., 0., 0.], [0., 0., 0., 4., 1.], [0., 0., 0., 2., -3.]]
B = [[1., 4.], [-3., -3.], [-2., 1.], [5., 2.], [-3., -3.]]
C = [[4., 2., -3., 2., -4.], [1., 4., 3., 0., 1.]]
D = [[1., 6.], [1., 0.]]
sys = self.sys1 + self.sys2
np.testing.assert_array_almost_equal(sys.A, A)
np.testing.assert_array_almost_equal(sys.B, B)
np.testing.assert_array_almost_equal(sys.C, C)
np.testing.assert_array_almost_equal(sys.D, D)
def testSub(self):
"""Subtract two MIMO systems."""
A = [[-3., 4., 2., 0., 0.], [-1., -3., 0., 0., 0.],
[2., 5., 3., 0., 0.], [0., 0., 0., 4., 1.], [0., 0., 0., 2., -3.]]
B = [[1., 4.], [-3., -3.], [-2., 1.], [5., 2.], [-3., -3.]]
C = [[4., 2., -3., -2., 4.], [1., 4., 3., 0., -1.]]
D = [[-5., 2.], [-1., 2.]]
sys = self.sys1 - self.sys2
np.testing.assert_array_almost_equal(sys.A, A)
np.testing.assert_array_almost_equal(sys.B, B)
np.testing.assert_array_almost_equal(sys.C, C)
np.testing.assert_array_almost_equal(sys.D, D)
def testMul(self):
"""Multiply two MIMO systems."""
A = [[4., 1., 0., 0., 0.], [2., -3., 0., 0., 0.],
[2., 0., -3., 4., 2.], [-6., 9., -1., -3., 0.],
[-4., 9., 2., 5., 3.]]
B = [[5., 2.], [-3., -3.], [7., -2.], [-12., -3.], [-5., -5.]]
C = [[-4., 12., 4., 2., -3.], [0., 1., 1., 4., 3.]]
D = [[-2., -8.], [1., -1.]]
sys = self.sys1 * self.sys2
np.testing.assert_array_almost_equal(sys.A, A)
np.testing.assert_array_almost_equal(sys.B, B)
np.testing.assert_array_almost_equal(sys.C, C)
np.testing.assert_array_almost_equal(sys.D, D)
def testEvalFr(self):
"""Evaluate the frequency response at one frequency."""
A = [[-2, 0.5], [0.5, -0.3]]
B = [[0.3, -1.3], [0.1, 0.]]
C = [[0., 0.1], [-0.3, -0.2]]
D = [[0., -0.8], [-0.3, 0.]]
sys = StateSpace(A, B, C, D)
resp = [[
4.37636761487965e-05 - 0.0152297592997812j,
-0.792603938730853 + 0.0261706783369803j
],
[
-0.331544857768052 + 0.0576105032822757j,
0.128919037199125 - 0.143824945295405j
]]
np.testing.assert_almost_equal(sys.evalfr(1.), resp)
def testFreqResp(self):
"""Evaluate the frequency response at multiple frequencies."""
A = [[-2, 0.5], [0.5, -0.3]]
B = [[0.3, -1.3], [0.1, 0.]]
C = [[0., 0.1], [-0.3, -0.2]]
D = [[0., -0.8], [-0.3, 0.]]
sys = StateSpace(A, B, C, D)
truemag = [[[0.0852992637230322, 0.00103596611395218],
[0.935374692849736, 0.799380720864549]],
[[0.55656854563842, 0.301542699860857],
[0.609178071542849, 0.0382108097985257]]]
truephase = [[[-0.566195599644593, -1.68063565332582],
[3.0465958317514, 3.14141384339534]],
[[2.90457947657161, 3.10601268291914],
[-0.438157380501337, -1.40720969147217]]]
trueomega = [0.1, 10.]
mag, phase, omega = sys.freqresp(trueomega)
np.testing.assert_almost_equal(mag, truemag)
np.testing.assert_almost_equal(phase, truephase)
np.testing.assert_equal(omega, trueomega)
def testMinreal(self):
"""Test a minreal model reduction"""
#A = [-2, 0.5, 0; 0.5, -0.3, 0; 0, 0, -0.1]
A = [[-2, 0.5, 0], [0.5, -0.3, 0], [0, 0, -0.1]]
#B = [0.3, -1.3; 0.1, 0; 1, 0]
B = [[0.3, -1.3], [0.1, 0.], [1.0, 0.0]]
#C = [0, 0.1, 0; -0.3, -0.2, 0]
C = [[0., 0.1, 0.0], [-0.3, -0.2, 0.0]]
#D = [0 -0.8; -0.3 0]
D = [[0., -0.8], [-0.3, 0.]]
# sys = ss(A, B, C, D)
sys = StateSpace(A, B, C, D)
sysr = sys.minreal()
self.assertEqual(sysr.states, 2)
self.assertEqual(sysr.inputs, sys.inputs)
self.assertEqual(sysr.outputs, sys.outputs)
np.testing.assert_array_almost_equal(eigvals(sysr.A),
[-2.136154, -0.1638459])
def testAppendSS(self):
"""Test appending two state-space systems"""
A1 = [[-2, 0.5, 0], [0.5, -0.3, 0], [0, 0, -0.1]]
B1 = [[0.3, -1.3], [0.1, 0.], [1.0, 0.0]]
C1 = [[0., 0.1, 0.0], [-0.3, -0.2, 0.0]]
D1 = [[0., -0.8], [-0.3, 0.]]
A2 = [[-1.]]
B2 = [[1.2]]
C2 = [[0.5]]
D2 = [[0.4]]
A3 = [[-2, 0.5, 0, 0], [0.5, -0.3, 0, 0], [0, 0, -0.1, 0],
[0, 0, 0., -1.]]
B3 = [[0.3, -1.3, 0], [0.1, 0., 0], [1.0, 0.0, 0], [0., 0, 1.2]]
C3 = [[0., 0.1, 0.0, 0.0], [-0.3, -0.2, 0.0, 0.0], [0., 0., 0., 0.5]]
D3 = [[0., -0.8, 0.], [-0.3, 0., 0.], [0., 0., 0.4]]
sys1 = StateSpace(A1, B1, C1, D1)
sys2 = StateSpace(A2, B2, C2, D2)
sys3 = StateSpace(A3, B3, C3, D3)
sys3c = sys1.append(sys2)
np.testing.assert_array_almost_equal(sys3.A, sys3c.A)
np.testing.assert_array_almost_equal(sys3.B, sys3c.B)
np.testing.assert_array_almost_equal(sys3.C, sys3c.C)
np.testing.assert_array_almost_equal(sys3.D, sys3c.D)
def testAppendTF(self):
"""Test appending a state-space system with a tf"""
A1 = [[-2, 0.5, 0], [0.5, -0.3, 0], [0, 0, -0.1]]
B1 = [[0.3, -1.3], [0.1, 0.], [1.0, 0.0]]
C1 = [[0., 0.1, 0.0], [-0.3, -0.2, 0.0]]
D1 = [[0., -0.8], [-0.3, 0.]]
s = TransferFunction([1, 0], [1])
h = 1 / (s + 1) / (s + 2)
sys1 = StateSpace(A1, B1, C1, D1)
sys2 = _convertToStateSpace(h)
sys3c = sys1.append(sys2)
np.testing.assert_array_almost_equal(sys1.A, sys3c.A[:3, :3])
np.testing.assert_array_almost_equal(sys1.B, sys3c.B[:3, :2])
np.testing.assert_array_almost_equal(sys1.C, sys3c.C[:2, :3])
np.testing.assert_array_almost_equal(sys1.D, sys3c.D[:2, :2])
np.testing.assert_array_almost_equal(sys2.A, sys3c.A[3:, 3:])
np.testing.assert_array_almost_equal(sys2.B, sys3c.B[3:, 2:])
np.testing.assert_array_almost_equal(sys2.C, sys3c.C[2:, 3:])
np.testing.assert_array_almost_equal(sys2.D, sys3c.D[2:, 2:])
np.testing.assert_array_almost_equal(sys3c.A[:3, 3:], np.zeros((3, 2)))
np.testing.assert_array_almost_equal(sys3c.A[3:, :3], np.zeros((2, 3)))
def testArrayAccessSS(self):
sys1 = StateSpace([[1., 2.], [3., 4.]], [[5., 6.], [6., 8.]],
[[9., 10.], [11., 12.]], [[13., 14.], [15., 16.]], 1)
sys1_11 = sys1[0, 1]
np.testing.assert_array_almost_equal(sys1_11.A, sys1.A)
np.testing.assert_array_almost_equal(sys1_11.B, sys1.B[:, 1])
np.testing.assert_array_almost_equal(sys1_11.C, sys1.C[0, :])
np.testing.assert_array_almost_equal(sys1_11.D, sys1.D[0, 1])
assert sys1.dt == sys1_11.dt
def test_dcgain(self):
sys = StateSpace(-2., 6., 5., 0)
np.testing.assert_equal(sys.dcgain(), 15.)
sys2 = StateSpace(-2, [6., 4.], [[5.], [7.], [11]], np.zeros((3, 2)))
expected = np.array([[15., 10.], [21., 14.], [33., 22.]])
np.testing.assert_array_equal(sys2.dcgain(), expected)
sys3 = StateSpace(0., 1., 1., 0.)
np.testing.assert_equal(sys3.dcgain(), np.nan)
class TestStateSpace(unittest.TestCase):
"""Tests for the StateSpace class."""
def setUp(self):
"""Set up a MIMO system to test operations on."""
A = [[-3., 4., 2.], [-1., -3., 0.], [2., 5., 3.]]
B = [[1., 4.], [-3., -3.], [-2., 1.]]
C = [[4., 2., -3.], [1., 4., 3.]]
D = [[-2., 4.], [0., 1.]]
a = [[4., 1.], [2., -3]]
b = [[5., 2.], [-3., -3.]]
c = [[2., -4], [0., 1.]]
d = [[3., 2.], [1., -1.]]
self.sys1 = StateSpace(A, B, C, D)
self.sys2 = StateSpace(a, b, c, d)
def testPole(self):
"""Evaluate the poles of a MIMO system."""
p = self.sys1.pole()
np.testing.assert_array_almost_equal(p, [3.34747678408874,
-3.17373839204437 + 1.47492908003839j,
-3.17373839204437 - 1.47492908003839j])
def testZero(self):
"""Evaluate the zeros of a SISO system."""
sys = StateSpace(self.sys1.A, [[3.], [-2.], [4.]], [[-1., 3., 2.]], [[-4.]])
z = sys.zero()
np.testing.assert_array_almost_equal(z, [4.26864638637134,
-3.75932319318567 + 1.10087776649554j,
-3.75932319318567 - 1.10087776649554j])
def testAdd(self):
"""Add two MIMO systems."""
A = [[-3., 4., 2., 0., 0.], [-1., -3., 0., 0., 0.],
[2., 5., 3., 0., 0.], [0., 0., 0., 4., 1.], [0., 0., 0., 2., -3.]]
B = [[1., 4.], [-3., -3.], [-2., 1.], [5., 2.], [-3., -3.]]
C = [[4., 2., -3., 2., -4.], [1., 4., 3., 0., 1.]]
D = [[1., 6.], [1., 0.]]
sys = self.sys1 + self.sys2
np.testing.assert_array_almost_equal(sys.A, A)
np.testing.assert_array_almost_equal(sys.B, B)
np.testing.assert_array_almost_equal(sys.C, C)
np.testing.assert_array_almost_equal(sys.D, D)
def testSub(self):
"""Subtract two MIMO systems."""
A = [[-3., 4., 2., 0., 0.], [-1., -3., 0., 0., 0.],
[2., 5., 3., 0., 0.], [0., 0., 0., 4., 1.], [0., 0., 0., 2., -3.]]
B = [[1., 4.], [-3., -3.], [-2., 1.], [5., 2.], [-3., -3.]]
C = [[4., 2., -3., -2., 4.], [1., 4., 3., 0., -1.]]
D = [[-5., 2.], [-1., 2.]]
sys = self.sys1 - self.sys2
np.testing.assert_array_almost_equal(sys.A, A)
np.testing.assert_array_almost_equal(sys.B, B)
np.testing.assert_array_almost_equal(sys.C, C)
np.testing.assert_array_almost_equal(sys.D, D)
def testMul(self):
"""Multiply two MIMO systems."""
A = [[4., 1., 0., 0., 0.], [2., -3., 0., 0., 0.], [2., 0., -3., 4., 2.],
[-6., 9., -1., -3., 0.], [-4., 9., 2., 5., 3.]]
B = [[5., 2.], [-3., -3.], [7., -2.], [-12., -3.], [-5., -5.]]
C = [[-4., 12., 4., 2., -3.], [0., 1., 1., 4., 3.]]
D = [[-2., -8.], [1., -1.]]
sys = self.sys1 * self.sys2
np.testing.assert_array_almost_equal(sys.A, A)
np.testing.assert_array_almost_equal(sys.B, B)
np.testing.assert_array_almost_equal(sys.C, C)
np.testing.assert_array_almost_equal(sys.D, D)
def testEvalFr(self):
"""Evaluate the frequency response at one frequency."""
A = [[-2, 0.5], [0.5, -0.3]]
B = [[0.3, -1.3], [0.1, 0.]]
C = [[0., 0.1], [-0.3, -0.2]]
D = [[0., -0.8], [-0.3, 0.]]
sys = StateSpace(A, B, C, D)
resp = [[4.37636761487965e-05 - 0.0152297592997812j,
-0.792603938730853 + 0.0261706783369803j],
[-0.331544857768052 + 0.0576105032822757j,
0.128919037199125 - 0.143824945295405j]]
np.testing.assert_almost_equal(sys.evalfr(1.), resp)
def testFreqResp(self):
"""Evaluate the frequency response at multiple frequencies."""
A = [[-2, 0.5], [0.5, -0.3]]
B = [[0.3, -1.3], [0.1, 0.]]
C = [[0., 0.1], [-0.3, -0.2]]
D = [[0., -0.8], [-0.3, 0.]]
sys = StateSpace(A, B, C, D)
truemag = [[[0.0852992637230322, 0.00103596611395218],
[0.935374692849736, 0.799380720864549]],
[[0.55656854563842, 0.301542699860857],
[0.609178071542849, 0.0382108097985257]]]
truephase = [[[-0.566195599644593, -1.68063565332582],
[3.0465958317514, 3.14141384339534]],
[[2.90457947657161, 3.10601268291914],
[-0.438157380501337, -1.40720969147217]]]
trueomega = [0.1, 10.]
mag, phase, omega = sys.freqresp(trueomega)
np.testing.assert_almost_equal(mag, truemag)
np.testing.assert_almost_equal(phase, truephase)
np.testing.assert_equal(omega, trueomega)
def testMinreal(self):
"""Test a minreal model reduction"""
#A = [-2, 0.5, 0; 0.5, -0.3, 0; 0, 0, -0.1]
A = [[-2, 0.5, 0], [0.5, -0.3, 0], [0, 0, -0.1]]
#B = [0.3, -1.3; 0.1, 0; 1, 0]
B = [[0.3, -1.3], [0.1, 0.], [1.0, 0.0]]
#C = [0, 0.1, 0; -0.3, -0.2, 0]
C = [[0., 0.1, 0.0], [-0.3, -0.2, 0.0]]
#D = [0 -0.8; -0.3 0]
D = [[0., -0.8], [-0.3, 0.]]
# sys = ss(A, B, C, D)
sys = StateSpace(A, B, C, D)
sysr = sys.minreal()
self.assertEqual(sysr.states, 2)
self.assertEqual(sysr.inputs, sys.inputs)
self.assertEqual(sysr.outputs, sys.outputs)
np.testing.assert_array_almost_equal(
eigvals(sysr.A), [-2.136154, -0.1638459])
def testAppendSS(self):
"""Test appending two state-space systems"""
A1 = [[-2, 0.5, 0], [0.5, -0.3, 0], [0, 0, -0.1]]
B1 = [[0.3, -1.3], [0.1, 0.], [1.0, 0.0]]
C1 = [[0., 0.1, 0.0], [-0.3, -0.2, 0.0]]
D1 = [[0., -0.8], [-0.3, 0.]]
A2 = [[-1.]]
B2 = [[1.2]]
C2 = [[0.5]]
D2 = [[0.4]]
A3 = [[-2, 0.5, 0, 0], [0.5, -0.3, 0, 0], [0, 0, -0.1, 0],
[0, 0, 0., -1.]]
B3 = [[0.3, -1.3, 0], [0.1, 0., 0], [1.0, 0.0, 0], [0., 0, 1.2]]
C3 = [[0., 0.1, 0.0, 0.0], [-0.3, -0.2, 0.0, 0.0], [0., 0., 0., 0.5]]
D3 = [[0., -0.8, 0.], [-0.3, 0., 0.], [0., 0., 0.4]]
sys1 = StateSpace(A1, B1, C1, D1)
sys2 = StateSpace(A2, B2, C2, D2)
sys3 = StateSpace(A3, B3, C3, D3)
sys3c = sys1.append(sys2)
np.testing.assert_array_almost_equal(sys3.A, sys3c.A)
np.testing.assert_array_almost_equal(sys3.B, sys3c.B)
np.testing.assert_array_almost_equal(sys3.C, sys3c.C)
np.testing.assert_array_almost_equal(sys3.D, sys3c.D)
def testAppendTF(self):
"""Test appending a state-space system with a tf"""
A1 = [[-2, 0.5, 0], [0.5, -0.3, 0], [0, 0, -0.1]]
B1 = [[0.3, -1.3], [0.1, 0.], [1.0, 0.0]]
C1 = [[0., 0.1, 0.0], [-0.3, -0.2, 0.0]]
D1 = [[0., -0.8], [-0.3, 0.]]
s = TransferFunction([1, 0], [1])
h = 1/(s+1)/(s+2)
sys1 = StateSpace(A1, B1, C1, D1)
sys2 = _convertToStateSpace(h)
sys3c = sys1.append(sys2)
np.testing.assert_array_almost_equal(sys1.A, sys3c.A[:3,:3])
np.testing.assert_array_almost_equal(sys1.B, sys3c.B[:3,:2])
np.testing.assert_array_almost_equal(sys1.C, sys3c.C[:2,:3])
np.testing.assert_array_almost_equal(sys1.D, sys3c.D[:2,:2])
np.testing.assert_array_almost_equal(sys2.A, sys3c.A[3:,3:])
np.testing.assert_array_almost_equal(sys2.B, sys3c.B[3:,2:])
np.testing.assert_array_almost_equal(sys2.C, sys3c.C[2:,3:])
np.testing.assert_array_almost_equal(sys2.D, sys3c.D[2:,2:])
np.testing.assert_array_almost_equal(sys3c.A[:3,3:], np.zeros( (3, 2)) )
np.testing.assert_array_almost_equal(sys3c.A[3:,:3], np.zeros( (2, 3)) )
def testArrayAccessSS(self):
sys1 = StateSpace([[1., 2.], [3., 4.]],
[[5., 6.], [6., 8.]],
[[9., 10.], [11., 12.]],
[[13., 14.], [15., 16.]], 1)
sys1_11 = sys1[0,1]
np.testing.assert_array_almost_equal(sys1_11.A,
sys1.A)
np.testing.assert_array_almost_equal(sys1_11.B,
sys1.B[:,1])
np.testing.assert_array_almost_equal(sys1_11.C,
sys1.C[0,:])
np.testing.assert_array_almost_equal(sys1_11.D,
sys1.D[0,1])
assert sys1.dt == sys1_11.dt
class TestStateSpace(unittest.TestCase):
"""Tests for the StateSpace class."""
def setUp(self):
"""Set up a MIMO system to test operations on."""
# sys1: 3-states square system (2 inputs x 2 outputs)
A322 = [[-3., 4., 2.],
[-1., -3., 0.],
[2., 5., 3.]]
B322 = [[1., 4.],
[-3., -3.],
[-2., 1.]]
C322 = [[4., 2., -3.],
[1., 4., 3.]]
D322 = [[-2., 4.],
[0., 1.]]
self.sys322 = StateSpace(A322, B322, C322, D322)
# sys1: 2-states square system (2 inputs x 2 outputs)
A222 = [[4., 1.],
[2., -3]]
B222 = [[5., 2.],
[-3., -3.]]
C222 = [[2., -4],
[0., 1.]]
D222 = [[3., 2.],
[1., -1.]]
self.sys222 = StateSpace(A222, B222, C222, D222)
# sys3: 6 states non square system (2 inputs x 3 outputs)
A623 = np.array([[1, 0, 0, 0, 0, 0],
[0, 1, 0, 0, 0, 0],
[0, 0, 3, 0, 0, 0],
[0, 0, 0, -4, 0, 0],
[0, 0, 0, 0, -1, 0],
[0, 0, 0, 0, 0, 3]])
B623 = np.array([[0, -1],
[-1, 0],
[1, -1],
[0, 0],
[0, 1],
[-1, -1]])
C623 = np.array([[1, 0, 0, 1, 0, 0],
[0, 1, 0, 1, 0, 1],
[0, 0, 1, 0, 0, 1]])
D623 = np.zeros((3, 2))
self.sys623 = StateSpace(A623, B623, C623, D623)
def test_pole(self):
"""Evaluate the poles of a MIMO system."""
p = np.sort(self.sys322.pole())
true_p = np.sort([3.34747678408874,
-3.17373839204437 + 1.47492908003839j,
-3.17373839204437 - 1.47492908003839j])
np.testing.assert_array_almost_equal(p, true_p)
def test_zero_empty(self):
"""Test to make sure zero() works with no zeros in system."""
sys = _convertToStateSpace(TransferFunction([1], [1, 2, 1]))
np.testing.assert_array_equal(sys.zero(), np.array([]))
@unittest.skipIf(not slycot_check(), "slycot not installed")
def test_zero_siso(self):
"""Evaluate the zeros of a SISO system."""
# extract only first input / first output system of sys222. This system is denoted sys111
# or tf111
tf111 = ss2tf(self.sys222)
sys111 = tf2ss(tf111[0, 0])
# compute zeros as root of the characteristic polynomial at the numerator of tf111
# this method is simple and assumed as valid in this test
true_z = np.sort(tf111[0, 0].zero())
# Compute the zeros through ab08nd, which is tested here
z = np.sort(sys111.zero())
np.testing.assert_almost_equal(true_z, z)
@unittest.skipIf(not slycot_check(), "slycot not installed")
def test_zero_mimo_sys322_square(self):
"""Evaluate the zeros of a square MIMO system."""
z = np.sort(self.sys322.zero())
true_z = np.sort([44.41465, -0.490252, -5.924398])
np.testing.assert_array_almost_equal(z, true_z)
@unittest.skipIf(not slycot_check(), "slycot not installed")
def test_zero_mimo_sys222_square(self):
"""Evaluate the zeros of a square MIMO system."""
z = np.sort(self.sys222.zero())
true_z = np.sort([-10.568501, 3.368501])
np.testing.assert_array_almost_equal(z, true_z)
@unittest.skipIf(not slycot_check(), "slycot not installed")
def test_zero_mimo_sys623_non_square(self):
"""Evaluate the zeros of a non square MIMO system."""
z = np.sort(self.sys623.zero())
true_z = np.sort([2., -1.])
np.testing.assert_array_almost_equal(z, true_z)
def test_add_ss(self):
"""Add two MIMO systems."""
A = [[-3., 4., 2., 0., 0.], [-1., -3., 0., 0., 0.],
[2., 5., 3., 0., 0.], [0., 0., 0., 4., 1.], [0., 0., 0., 2., -3.]]
B = [[1., 4.], [-3., -3.], [-2., 1.], [5., 2.], [-3., -3.]]
C = [[4., 2., -3., 2., -4.], [1., 4., 3., 0., 1.]]
D = [[1., 6.], [1., 0.]]
sys = self.sys322 + self.sys222
np.testing.assert_array_almost_equal(sys.A, A)
np.testing.assert_array_almost_equal(sys.B, B)
np.testing.assert_array_almost_equal(sys.C, C)
np.testing.assert_array_almost_equal(sys.D, D)
def test_subtract_ss(self):
"""Subtract two MIMO systems."""
A = [[-3., 4., 2., 0., 0.], [-1., -3., 0., 0., 0.],
[2., 5., 3., 0., 0.], [0., 0., 0., 4., 1.], [0., 0., 0., 2., -3.]]
B = [[1., 4.], [-3., -3.], [-2., 1.], [5., 2.], [-3., -3.]]
C = [[4., 2., -3., -2., 4.], [1., 4., 3., 0., -1.]]
D = [[-5., 2.], [-1., 2.]]
sys = self.sys322 - self.sys222
np.testing.assert_array_almost_equal(sys.A, A)
np.testing.assert_array_almost_equal(sys.B, B)
np.testing.assert_array_almost_equal(sys.C, C)
np.testing.assert_array_almost_equal(sys.D, D)
def test_multiply_ss(self):
"""Multiply two MIMO systems."""
A = [[4., 1., 0., 0., 0.], [2., -3., 0., 0., 0.], [2., 0., -3., 4., 2.],
[-6., 9., -1., -3., 0.], [-4., 9., 2., 5., 3.]]
B = [[5., 2.], [-3., -3.], [7., -2.], [-12., -3.], [-5., -5.]]
C = [[-4., 12., 4., 2., -3.], [0., 1., 1., 4., 3.]]
D = [[-2., -8.], [1., -1.]]
sys = self.sys322 * self.sys222
np.testing.assert_array_almost_equal(sys.A, A)
np.testing.assert_array_almost_equal(sys.B, B)
np.testing.assert_array_almost_equal(sys.C, C)
np.testing.assert_array_almost_equal(sys.D, D)
def test_evalfr(self):
"""Evaluate the frequency response at one frequency."""
A = [[-2, 0.5], [0.5, -0.3]]
B = [[0.3, -1.3], [0.1, 0.]]
C = [[0., 0.1], [-0.3, -0.2]]
D = [[0., -0.8], [-0.3, 0.]]
sys = StateSpace(A, B, C, D)
resp = [[4.37636761487965e-05 - 0.0152297592997812j,
-0.792603938730853 + 0.0261706783369803j],
[-0.331544857768052 + 0.0576105032822757j,
0.128919037199125 - 0.143824945295405j]]
# Correct versions of the call
np.testing.assert_almost_equal(evalfr(sys, 1j), resp)
np.testing.assert_almost_equal(sys._evalfr(1.), resp)
# Deprecated version of the call (should generate warning)
import warnings
with warnings.catch_warnings(record=True) as w:
# Set up warnings filter to only show warnings in control module
warnings.filterwarnings("ignore")
warnings.filterwarnings("always", module="control")
# Make sure that we get a pending deprecation warning
sys.evalfr(1.)
assert len(w) == 1
assert issubclass(w[-1].category, PendingDeprecationWarning)
@unittest.skipIf(not slycot_check(), "slycot not installed")
def test_freq_resp(self):
"""Evaluate the frequency response at multiple frequencies."""
A = [[-2, 0.5], [0.5, -0.3]]
B = [[0.3, -1.3], [0.1, 0.]]
C = [[0., 0.1], [-0.3, -0.2]]
D = [[0., -0.8], [-0.3, 0.]]
sys = StateSpace(A, B, C, D)
true_mag = [[[0.0852992637230322, 0.00103596611395218],
[0.935374692849736, 0.799380720864549]],
[[0.55656854563842, 0.301542699860857],
[0.609178071542849, 0.0382108097985257]]]
true_phase = [[[-0.566195599644593, -1.68063565332582],
[3.0465958317514, 3.14141384339534]],
[[2.90457947657161, 3.10601268291914],
[-0.438157380501337, -1.40720969147217]]]
true_omega = [0.1, 10.]
mag, phase, omega = sys.freqresp(true_omega)
np.testing.assert_almost_equal(mag, true_mag)
np.testing.assert_almost_equal(phase, true_phase)
np.testing.assert_equal(omega, true_omega)
@unittest.skipIf(not slycot_check(), "slycot not installed")
def test_minreal(self):
"""Test a minreal model reduction."""
# A = [-2, 0.5, 0; 0.5, -0.3, 0; 0, 0, -0.1]
A = [[-2, 0.5, 0], [0.5, -0.3, 0], [0, 0, -0.1]]
# B = [0.3, -1.3; 0.1, 0; 1, 0]
B = [[0.3, -1.3], [0.1, 0.], [1.0, 0.0]]
# C = [0, 0.1, 0; -0.3, -0.2, 0]
C = [[0., 0.1, 0.0], [-0.3, -0.2, 0.0]]
# D = [0 -0.8; -0.3 0]
D = [[0., -0.8], [-0.3, 0.]]
# sys = ss(A, B, C, D)
sys = StateSpace(A, B, C, D)
sysr = sys.minreal()
self.assertEqual(sysr.states, 2)
self.assertEqual(sysr.inputs, sys.inputs)
self.assertEqual(sysr.outputs, sys.outputs)
np.testing.assert_array_almost_equal(
eigvals(sysr.A), [-2.136154, -0.1638459])
def test_append_ss(self):
"""Test appending two state-space systems."""
A1 = [[-2, 0.5, 0], [0.5, -0.3, 0], [0, 0, -0.1]]
B1 = [[0.3, -1.3], [0.1, 0.], [1.0, 0.0]]
C1 = [[0., 0.1, 0.0], [-0.3, -0.2, 0.0]]
D1 = [[0., -0.8], [-0.3, 0.]]
A2 = [[-1.]]
B2 = [[1.2]]
C2 = [[0.5]]
D2 = [[0.4]]
A3 = [[-2, 0.5, 0, 0], [0.5, -0.3, 0, 0], [0, 0, -0.1, 0],
[0, 0, 0., -1.]]
B3 = [[0.3, -1.3, 0], [0.1, 0., 0], [1.0, 0.0, 0], [0., 0, 1.2]]
C3 = [[0., 0.1, 0.0, 0.0], [-0.3, -0.2, 0.0, 0.0], [0., 0., 0., 0.5]]
D3 = [[0., -0.8, 0.], [-0.3, 0., 0.], [0., 0., 0.4]]
sys1 = StateSpace(A1, B1, C1, D1)
sys2 = StateSpace(A2, B2, C2, D2)
sys3 = StateSpace(A3, B3, C3, D3)
sys3c = sys1.append(sys2)
np.testing.assert_array_almost_equal(sys3.A, sys3c.A)
np.testing.assert_array_almost_equal(sys3.B, sys3c.B)
np.testing.assert_array_almost_equal(sys3.C, sys3c.C)
np.testing.assert_array_almost_equal(sys3.D, sys3c.D)
def test_append_tf(self):
"""Test appending a state-space system with a tf"""
A1 = [[-2, 0.5, 0], [0.5, -0.3, 0], [0, 0, -0.1]]
B1 = [[0.3, -1.3], [0.1, 0.], [1.0, 0.0]]
C1 = [[0., 0.1, 0.0], [-0.3, -0.2, 0.0]]
D1 = [[0., -0.8], [-0.3, 0.]]
s = TransferFunction([1, 0], [1])
h = 1 / (s + 1) / (s + 2)
sys1 = StateSpace(A1, B1, C1, D1)
sys2 = _convertToStateSpace(h)
sys3c = sys1.append(sys2)
np.testing.assert_array_almost_equal(sys1.A, sys3c.A[:3, :3])
np.testing.assert_array_almost_equal(sys1.B, sys3c.B[:3, :2])
np.testing.assert_array_almost_equal(sys1.C, sys3c.C[:2, :3])
np.testing.assert_array_almost_equal(sys1.D, sys3c.D[:2, :2])
np.testing.assert_array_almost_equal(sys2.A, sys3c.A[3:, 3:])
np.testing.assert_array_almost_equal(sys2.B, sys3c.B[3:, 2:])
np.testing.assert_array_almost_equal(sys2.C, sys3c.C[2:, 3:])
np.testing.assert_array_almost_equal(sys2.D, sys3c.D[2:, 2:])
np.testing.assert_array_almost_equal(sys3c.A[:3, 3:], np.zeros((3, 2)))
np.testing.assert_array_almost_equal(sys3c.A[3:, :3], np.zeros((2, 3)))
def test_array_access_ss(self):
sys1 = StateSpace([[1., 2.], [3., 4.]],
[[5., 6.], [6., 8.]],
[[9., 10.], [11., 12.]],
[[13., 14.], [15., 16.]], 1)
sys1_11 = sys1[0, 1]
np.testing.assert_array_almost_equal(sys1_11.A,
sys1.A)
np.testing.assert_array_almost_equal(sys1_11.B,
sys1.B[:, 1])
np.testing.assert_array_almost_equal(sys1_11.C,
sys1.C[0, :])
np.testing.assert_array_almost_equal(sys1_11.D,
sys1.D[0, 1])
assert sys1.dt == sys1_11.dt
def test_dc_gain_cont(self):
"""Test DC gain for continuous-time state-space systems."""
sys = StateSpace(-2., 6., 5., 0)
np.testing.assert_equal(sys.dcgain(), 15.)
sys2 = StateSpace(-2, [6., 4.], [[5.], [7.], [11]], np.zeros((3, 2)))
expected = np.array([[15., 10.], [21., 14.], [33., 22.]])
np.testing.assert_array_equal(sys2.dcgain(), expected)
sys3 = StateSpace(0., 1., 1., 0.)
np.testing.assert_equal(sys3.dcgain(), np.nan)
def test_dc_gain_discr(self):
"""Test DC gain for discrete-time state-space systems."""
# static gain
sys = StateSpace([], [], [], 2, True)
np.testing.assert_equal(sys.dcgain(), 2)
# averaging filter
sys = StateSpace(0.5, 0.5, 1, 0, True)
np.testing.assert_almost_equal(sys.dcgain(), 1)
# differencer
sys = StateSpace(0, 1, -1, 1, True)
np.testing.assert_equal(sys.dcgain(), 0)
# summer
sys = StateSpace(1, 1, 1, 0, True)
np.testing.assert_equal(sys.dcgain(), np.nan)
def test_dc_gain_integrator(self):
"""DC gain when eigenvalue at DC returns appropriately sized array of nan."""
# the SISO case is also tested in test_dc_gain_{cont,discr}
import itertools
# iterate over input and output sizes, and continuous (dt=None) and discrete (dt=True) time
for inputs, outputs, dt in itertools.product(range(1, 6), range(1, 6), [None, True]):
states = max(inputs, outputs)
# a matrix that is singular at DC, and has no "useless" states as in
# _remove_useless_states
a = np.triu(np.tile(2, (states, states)))
# eigenvalues all +2, except for ...
a[0, 0] = 0 if dt is None else 1
b = np.eye(max(inputs, states))[:states, :inputs]
c = np.eye(max(outputs, states))[:outputs, :states]
d = np.zeros((outputs, inputs))
sys = StateSpace(a, b, c, d, dt)
dc = np.squeeze(np.tile(np.nan, (outputs, inputs)))
np.testing.assert_array_equal(dc, sys.dcgain())
def test_scalar_static_gain(self):
"""Regression: can we create a scalar static gain?"""
g1 = StateSpace([], [], [], [2])
g2 = StateSpace([], [], [], [3])
# make sure StateSpace internals, specifically ABC matrix
# sizes, are OK for LTI operations
g3 = g1 * g2
self.assertEqual(6, g3.D[0, 0])
g4 = g1 + g2
self.assertEqual(5, g4.D[0, 0])
g5 = g1.feedback(g2)
self.assertAlmostEqual(2. / 7, g5.D[0, 0])
g6 = g1.append(g2)
np.testing.assert_array_equal(np.diag([2, 3]), g6.D)
def test_matrix_static_gain(self):
"""Regression: can we create matrix static gains?"""
d1 = np.matrix([[1, 2, 3], [4, 5, 6]])
d2 = np.matrix([[7, 8], [9, 10], [11, 12]])
g1 = StateSpace([], [], [], d1)
# _remove_useless_states was making A = [[0]]
self.assertEqual((0, 0), g1.A.shape)
g2 = StateSpace([], [], [], d2)
g3 = StateSpace([], [], [], d2.T)
h1 = g1 * g2
np.testing.assert_array_equal(d1 * d2, h1.D)
h2 = g1 + g3
np.testing.assert_array_equal(d1 + d2.T, h2.D)
h3 = g1.feedback(g2)
np.testing.assert_array_almost_equal(
solve(np.eye(2) + d1 * d2, d1), h3.D)
h4 = g1.append(g2)
np.testing.assert_array_equal(block_diag(d1, d2), h4.D)
def test_remove_useless_states(self):
"""Regression: _remove_useless_states gives correct ABC sizes."""
g1 = StateSpace(np.zeros((3, 3)),
np.zeros((3, 4)),
np.zeros((5, 3)),
np.zeros((5, 4)))
self.assertEqual((0, 0), g1.A.shape)
self.assertEqual((0, 4), g1.B.shape)
self.assertEqual((5, 0), g1.C.shape)
self.assertEqual((5, 4), g1.D.shape)
self.assertEqual(0, g1.states)
def test_bad_empty_matrices(self):
"""Mismatched ABCD matrices when some are empty."""
self.assertRaises(ValueError, StateSpace, [1], [], [], [1])
self.assertRaises(ValueError, StateSpace, [1], [1], [], [1])
self.assertRaises(ValueError, StateSpace, [1], [], [1], [1])
self.assertRaises(ValueError, StateSpace, [], [1], [], [1])
self.assertRaises(ValueError, StateSpace, [], [1], [1], [1])
self.assertRaises(ValueError, StateSpace, [], [], [1], [1])
self.assertRaises(ValueError, StateSpace, [1], [1], [1], [])
def test_minreal_static_gain(self):
"""Regression: minreal on static gain was failing."""
g1 = StateSpace([], [], [], [1])
g2 = g1.minreal()
np.testing.assert_array_equal(g1.A, g2.A)
np.testing.assert_array_equal(g1.B, g2.B)
np.testing.assert_array_equal(g1.C, g2.C)
np.testing.assert_array_equal(g1.D, g2.D)
def test_empty(self):
"""Regression: can we create an empty StateSpace object?"""
g1 = StateSpace([], [], [], [])
self.assertEqual(0, g1.states)
self.assertEqual(0, g1.inputs)
self.assertEqual(0, g1.outputs)
def test_matrix_to_state_space(self):
"""_convertToStateSpace(matrix) gives ss([],[],[],D)"""
D = np.matrix([[1, 2, 3], [4, 5, 6]])
g = _convertToStateSpace(D)
def empty(shape):
m = np.matrix([])
m.shape = shape
return m
np.testing.assert_array_equal(empty((0, 0)), g.A)
np.testing.assert_array_equal(empty((0, D.shape[1])), g.B)
np.testing.assert_array_equal(empty((D.shape[0], 0)), g.C)
np.testing.assert_array_equal(D, g.D)
def test_lft(self):
""" test lft function with result obtained from matlab implementation"""
# test case
A = [[1, 2, 3],
[1, 4, 5],
[2, 3, 4]]
B = [[0, 2],
[5, 6],
[5, 2]]
C = [[1, 4, 5],
[2, 3, 0]]
D = [[0, 0],
[3, 0]]
P = StateSpace(A, B, C, D)
Ak = [[0, 2, 3],
[2, 3, 5],
[2, 1, 9]]
Bk = [[1, 1],
[2, 3],
[9, 4]]
Ck = [[1, 4, 5],
[2, 3, 6]]
Dk = [[0, 2],
[0, 0]]
K = StateSpace(Ak, Bk, Ck, Dk)
# case 1
pk = P.lft(K, 2, 1)
Amatlab = [1, 2, 3, 4, 6, 12, 1, 4, 5, 17, 38, 61, 2, 3, 4, 9, 26, 37, 2, 3, 0, 3, 14, 18, 4, 6, 0, 8, 27, 35, 18, 27, 0, 29, 109, 144]
Bmatlab = [0, 10, 10, 7, 15, 58]
Cmatlab = [1, 4, 5, 0, 0, 0]
Dmatlab = [0]
np.testing.assert_allclose(np.array(pk.A).reshape(-1), Amatlab)
np.testing.assert_allclose(np.array(pk.B).reshape(-1), Bmatlab)
np.testing.assert_allclose(np.array(pk.C).reshape(-1), Cmatlab)
np.testing.assert_allclose(np.array(pk.D).reshape(-1), Dmatlab)
# case 2
pk = P.lft(K)
Amatlab = [1, 2, 3, 4, 6, 12, -3, -2, 5, 11, 14, 31, -2, -3, 4, 3, 2, 7, 0.6, 3.4, 5, -0.6, -0.4, 0, 0.8, 6.2, 10, 0.2, -4.2, -4, 7.4, 33.6, 45, -0.4, -8.6, -3]
Bmatlab = []
Cmatlab = []
Dmatlab = []
np.testing.assert_allclose(np.array(pk.A).reshape(-1), Amatlab)
np.testing.assert_allclose(np.array(pk.B).reshape(-1), Bmatlab)
np.testing.assert_allclose(np.array(pk.C).reshape(-1), Cmatlab)
np.testing.assert_allclose(np.array(pk.D).reshape(-1), Dmatlab)
class TestStateSpace(unittest.TestCase):
"""Tests for the StateSpace class."""
def setUp(self):
"""Set up a MIMO system to test operations on."""
A = [[-3., 4., 2.], [-1., -3., 0.], [2., 5., 3.]]
B = [[1., 4.], [-3., -3.], [-2., 1.]]
C = [[4., 2., -3.], [1., 4., 3.]]
D = [[-2., 4.], [0., 1.]]
a = [[4., 1.], [2., -3]]
b = [[5., 2.], [-3., -3.]]
c = [[2., -4], [0., 1.]]
d = [[3., 2.], [1., -1.]]
self.sys1 = StateSpace(A, B, C, D)
self.sys2 = StateSpace(a, b, c, d)
def testPole(self):
"""Evaluate the poles of a MIMO system."""
p = self.sys1.pole()
np.testing.assert_array_almost_equal(p, [3.34747678408874,
-3.17373839204437 + 1.47492908003839j,
-3.17373839204437 - 1.47492908003839j])
def testZero(self):
"""Evaluate the zeros of a SISO system."""
sys = StateSpace(self.sys1.A, [[3.], [-2.], [4.]], [[-1., 3., 2.]], [[-4.]])
z = sys.zero()
np.testing.assert_array_almost_equal(z, [4.26864638637134,
-3.75932319318567 + 1.10087776649554j,
-3.75932319318567 - 1.10087776649554j])
def testAdd(self):
"""Add two MIMO systems."""
A = [[-3., 4., 2., 0., 0.], [-1., -3., 0., 0., 0.],
[2., 5., 3., 0., 0.], [0., 0., 0., 4., 1.], [0., 0., 0., 2., -3.]]
B = [[1., 4.], [-3., -3.], [-2., 1.], [5., 2.], [-3., -3.]]
C = [[4., 2., -3., 2., -4.], [1., 4., 3., 0., 1.]]
D = [[1., 6.], [1., 0.]]
sys = self.sys1 + self.sys2
np.testing.assert_array_almost_equal(sys.A, A)
np.testing.assert_array_almost_equal(sys.B, B)
np.testing.assert_array_almost_equal(sys.C, C)
np.testing.assert_array_almost_equal(sys.D, D)
def testSub(self):
"""Subtract two MIMO systems."""
A = [[-3., 4., 2., 0., 0.], [-1., -3., 0., 0., 0.],
[2., 5., 3., 0., 0.], [0., 0., 0., 4., 1.], [0., 0., 0., 2., -3.]]
B = [[1., 4.], [-3., -3.], [-2., 1.], [5., 2.], [-3., -3.]]
C = [[4., 2., -3., -2., 4.], [1., 4., 3., 0., -1.]]
D = [[-5., 2.], [-1., 2.]]
sys = self.sys1 - self.sys2
np.testing.assert_array_almost_equal(sys.A, A)
np.testing.assert_array_almost_equal(sys.B, B)
np.testing.assert_array_almost_equal(sys.C, C)
np.testing.assert_array_almost_equal(sys.D, D)
def testMul(self):
"""Multiply two MIMO systems."""
A = [[4., 1., 0., 0., 0.], [2., -3., 0., 0., 0.], [2., 0., -3., 4., 2.],
[-6., 9., -1., -3., 0.], [-4., 9., 2., 5., 3.]]
B = [[5., 2.], [-3., -3.], [7., -2.], [-12., -3.], [-5., -5.]]
C = [[-4., 12., 4., 2., -3.], [0., 1., 1., 4., 3.]]
D = [[-2., -8.], [1., -1.]]
sys = self.sys1 * self.sys2
np.testing.assert_array_almost_equal(sys.A, A)
np.testing.assert_array_almost_equal(sys.B, B)
np.testing.assert_array_almost_equal(sys.C, C)
np.testing.assert_array_almost_equal(sys.D, D)
def testEvalFr(self):
"""Evaluate the frequency response at one frequency."""
A = [[-2, 0.5], [0.5, -0.3]]
B = [[0.3, -1.3], [0.1, 0.]]
C = [[0., 0.1], [-0.3, -0.2]]
D = [[0., -0.8], [-0.3, 0.]]
sys = StateSpace(A, B, C, D)
resp = [[4.37636761487965e-05 - 0.0152297592997812j,
-0.792603938730853 + 0.0261706783369803j],
[-0.331544857768052 + 0.0576105032822757j,
0.128919037199125 - 0.143824945295405j]]
np.testing.assert_almost_equal(sys.evalfr(1.), resp)
def testFreqResp(self):
"""Evaluate the frequency response at multiple frequencies."""
A = [[-2, 0.5], [0.5, -0.3]]
B = [[0.3, -1.3], [0.1, 0.]]
C = [[0., 0.1], [-0.3, -0.2]]
D = [[0., -0.8], [-0.3, 0.]]
sys = StateSpace(A, B, C, D)
truemag = [[[0.0852992637230322, 0.00103596611395218],
[0.935374692849736, 0.799380720864549]],
[[0.55656854563842, 0.301542699860857],
[0.609178071542849, 0.0382108097985257]]]
truephase = [[[-0.566195599644593, -1.68063565332582],
[3.0465958317514, 3.14141384339534]],
[[2.90457947657161, 3.10601268291914],
[-0.438157380501337, -1.40720969147217]]]
trueomega = [0.1, 10.]
mag, phase, omega = sys.freqresp(trueomega)
np.testing.assert_almost_equal(mag, truemag)
np.testing.assert_almost_equal(phase, truephase)
np.testing.assert_equal(omega, trueomega)
